On Hausdorff dimension of radial projections

Abstract

For any $x\in\mathbb{R}^d$, $d\geq 2$, denote $\pi^x: \mathbb{R}^d\backslash\{x\}\rightarrow S^{d-1}$ as the radial projection $$\pi^x(y)=\frac{y-x}{|y-x|}. $$ Given a Borel set $E\subset{\Bbb R}^d$, $\dim_{\mathcal{H}} E\leq d-1$, in this paper we investigate for how many $x\in \mathbb{R}^d$ the radial projection $\pi^x$ preserves the Hausdorff dimension of $E$, namely whether $\dim_{\mathcal{H}}\pi^x(E)=\dim_{\mathcal{H}} E$. We develop a general framework to link $\pi^x(E)$, $x\in F$ and $\pi^y(F)$, $y\in E$, for any Borel set $F\subset\mathbb{R}^d$. In particular, whether $\dim_{\mathcal{H}}\pi^x(E)=\dim_{\mathcal{H}}E$ for some $x\in F$ can be reduced to whether $F$ is visible from some $y\in E$ (i.e. $\mathcal{H}^{d-1}(\pi^y(F))>0$). This allows us to apply Orponen's estimate on visibility to obtain $$\dim_{\mathcal{H}}\left\{x\in\mathbb{R}^d: \dim_{\mathcal{H}}\pi^x(E)<\dim_{\mathcal{H}}E\right\}\leq 2(d-1)-\dim_{\mathcal{H}}E,$$ for any Borel set $E\subset{\Bbb R}^d$, $\dim_{\mathcal{H}} E\in(d-2, d-1]$. This improves the Peres-Schlag bound when $\dim_{\mathcal{H}} E\in(d-\frac{3}{2}, d-1]$, and it is optimal at the endpoint $\dim_{\mathcal{H}} E=d-1$.